Isotopy types of knots of codimension two

Abstract

In this paper the classification of n-dimensional knots in S+2, bound- ing /--connected manifolds, where 3r>n + l>6, in terms of stable homotopy theory is suggested. The problem of isotopy classification is the fundamental one of knot theory. It was solved by A. Haefliger and J. Levine for knots of codimension > 3. The first step in reduction of a classification of knots of codimension two to a homotopy problem was made by R. Lashof and J. Shaneson (6). They showed that in the class of «-dimensional knots with group Z for n > 5 there are at most two different knots having homotopy equivalent exteriors. In 1970 J. Levine (10) gave an algebraic classification of (2q — l)-dimensional knots in S2q+X bounding (q — 1)- connected manifolds. It turned out that the only invariant determining the isotopy type of such knots is the Seifert matrix considered to within 5-equivalence. Later Trotter (16) and C. Kearton (5) obtained a classification of knots, studied by J. Levine, in terms of Blanchfield pairing. C. Kearton has partially analyzed the more difficult problem of classification of 2^r-dimensional knots in S2q+2 bounding (q — l)-connected manifolds. In the present paper the classification of a wider class of higher-dimensional knots is obtained. It is the class of «-dimensional knots in S+2 bounding r-connected manifolds, where 3r > n + 1 > 6. The main result of this paper is the construction of a one-to-one correspondence between the set of isotopy types of such knots and some set described in purely homotopic terms. The plan of the paper is as follows: In §1 submanifolds of the sphere of codimension one are considered. Such a submanifold is assigned some pairing in the sense of the Spanier-Whitehead theory. This pairing describes the homotopy linking of the submanifold with its copy translated in the direction of the positive normal field. This pairing induces the usual Seifert pairing on middle dimensional homology groups and is called the homotopy Seifert pairing. The main result of § 1 is the construction of a one-to-one correspondence between the set of isotopy types of such submanifolds and the set of homotopy pairings.